The decimal 511 is equal to 2⁹ − 1, which is one less than 512 and corresponds to nine ones in binary. In octal, this boundary value is represented as 777. 777₈ expands to 7×8² + 7×8¹ + 7×8⁰ = 448 + 56 + 7 = 511. Thus 777 is the correct octal representation of 511.
Option A:
757₈ equals 7×64 + 5×8 + 7 = 448 + 40 + 7 = 495. This is less than 511, so it does not match the given decimal number.
Option B:
767₈ represents 7×64 + 6×8 + 7 = 448 + 48 + 7 = 503. Because 503 differs from 511, this octal number is not correct.
Option C:
775₈ stands for 7×64 + 7×8 + 5 = 448 + 56 + 5 = 509. This value is close to but still less than 511, so it cannot be the exact representation.
Option D:
777₈ expands to 448 + 56 + 7 = 511, exactly equal to the decimal number specified. Its all-7 pattern reflects the maximum value for three octal digits, analogous to all ones in binary.
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